A sharp upper bound on the spectral radius of weighted graphs

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain an upper bound on the spectral radius of the adjacency matrix and characterize graphs for which the bound is attained.     

On an upper bound of the spectral radius of graphs

We give some upper bounds for the spectral radius of bipartite graph and graph, which improve the result in Hong’s Paper [Y. Hong, J.-L. Shu, K. Fang, A sharp upper bound of the spectral radius of graphs, J. Combin. Theory Ser. B 81 (2001) 177-183].     

A note on the spectral characterization of dumbbell graphs

The dumbbell graph, denoted by D_a,b,c, is a bicyclic graph consisting of two vertex-disjoint cycles C_a and C_b joined by a path P_c+3 (c-1) having only its end-vertices in common with the two cycles. By using a new cospectral invariant for (r,r+1)-almost regular graphs, we will show that almost all dumbbell graphs (without cycle C₄ as a subgraph) are determined by the adjacency spectrum.     

Perturbations on bidegreed graphs minimizing the spectral radius

One common problem in spectral graph theory is to determine which graphs, under some prescribed constraints, maximize or minimize the spectral radius of the adjacency matrix. Here we consider minimizers in the set of bidegreed, or biregular, graphs with pendant vertices and given degree sequence. In this setting, we consider a particular graph perturbation whose effect is to decrease the spectral radius. Hence we restrict the structure of minimizers for k-cyclic degree sequences.     

Improved upper and lower bounds for the spectral radius of digraphs

Let G be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius ρ(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, sharp upper and lower bounds on ρ(G) are given. We show that some known bounds can be obtained from our bounds.     

A sharp upper bound on the signless Laplacian spectral radius of graphs

Let G be a simple connected graph of order n   with degree sequence _d1,_d2,…,_dn$d_1,d_2,\dots ,d_n$ in non-increasing order. The signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ of G   is the largest eigenvalue of its signless Laplacian matrix Q(G)$Q(G)$. In this paper, we give a sharp upper bound on the signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ in terms of _di$d_i$, which improves and generalizes some known results.     

Sharp upper and lower bounds for the Laplacian spectral radius and the spectral radius of graphs

In this paper, sharp upper bounds for the Laplacian spectral radius and the spectral radius of graphs are given, respectively. We show that some known bounds can be obtained from our bounds. For a bipartite graph G, we also present sharp lower bounds for the Laplacian spectral radius and the spectral radius, respectively.     

An improved upper bound for the Laplacian spectral radius of graphs

Let G be a simple graph with n vertices, m edges. Let Δ and δ be the maximum and minimum degree of G, respectively. If each edge of G belongs to t triangles (t≥1), then we present a new upper bound for the Laplacian spectral radius of G as follows:

Moreover, we give an example to illustrate that our result is, in some cases, the best.     

双圈图的Laplace矩阵的谱半径

利用奇异点对的分类,得到了n阶双圈图的Laplace矩阵的谱半径的第二至第八大值,并且刻划了达到这些上界的极图.     

Halin图谱半径的新上界及极图

利用移接变形的方法再结合特征值的计算技巧刻画出Halin图中谱半径达到第二大的极图,从而得到除轮图以外的Halin图的谱半径的上界以及极图.     

The new upper bounds on the spectral radius of weighted graphs

Let us consider weighted graphs, where the weights of the edges are positive definite matrices. The eigenvalues of a weighted graph are the eigenvalues of its adjacency matrix and the spectral radius of a weighted graph is also the spectral radius of its adjacency matrix. In this paper, we obtain two upper bounds for the spectral radius of weighted graphs and compare with a known upper bound. We also characterize graphs for which the upper bounds are attained.     

The Laplacian spectral radius of graphs on surfaces

Let G be an n-vertex (n?3) simple graph embeddable on a surface of Euler genus γ (the number of crosscaps plus twice the number of handles). Denote by Δ the maximum degree of G. In this paper, we first present two upper bounds on the Laplacian spectral radius of G as follows:

(i)